To know when to use much or many depends on if it refers something countable or uncountable.
In this specific example people is countable, so it's correct to use many.
Also, using "The amount of people" might be correct grammar but it does sound slightly strange or awkward.
According to Dictionary.com Unabridged
Based on the Random House Dictionary, © Random House, Inc. 2017.
The traditional distinction between amount and number is that amount is used with mass or uncountable nouns ( the amount of paperwork; the amount of energy) and number with countable nouns ( a number of songs; a number of days).
Although objected to, the use of amount instead of number with countable nouns occurs in both speech and writing, especially when the noun can be considered as a unit or group ( the amount of people present; the amount of weapons) or when it refers to money ( the amount of dollars paid; the amount of pennies in the till).